Set (mathematics) - Special Sets

Special Sets

There are some sets which hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted {} or ∅. Another is the unit set {x} which contains exactly one element, namely x. Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include:

• P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
• N or ℕ, denoting the set of all natural numbers: N = {1, 2, 3, . . .} (sometimes defined containing 0).
• Z or ℤ, denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
• R or ℝ, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as √2, as well as transcendental numbers such as π, e and numbers that cannot be defined).
• C or ℂ, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.
• H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.

Positive and negative sets are denoted by a superscript - or +, for example: ℚ+ represents the set of positive rational numbers.

Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related fields.